(OT) mathematicians-discover-prime-conspiracy

[Antony Prince]

On March 25, 2016 12:34:51 AM EDT, Antony Prince <antony at blazrsoft.com> wrote:
>On March 24, 2016 11:17:58 PM EDT, "Marcio Barbado, Jr."
><marcio.barbado at gmail.com> wrote:
>>Not sure if it's counterintuitive once tossing can be seen as
>>abandoning inertia.
>>
>>
>>Marcio Barbado, Jr.
>>
>>
>>
>>On Fri, Mar 18, 2016 at 9:18 AM, Peter Lebbing
>><peter at digitalbrains.com> wrote:
>>> On 14/03/16 10:37, Fulano Diego Perez wrote:
>>>>
>>https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/
>>>
>>> So forgive me for the off-topicness, but something in the text
>caught
>>my
>>> attention:
>>>
>>>> Soundararajan was drawn to study consecutive primes after hearing a
>>>> lecture at Stanford by the mathematician Tadashi Tokieda, of the
>>>> University of Cambridge, in which he mentioned a counterintuitive
>>>> property of coin-tossing: If Alice tosses a coin until she sees a
>>>> head followed by a tail, and Bob tosses a coin until he sees two
>>>> heads in a row, then on average, Alice will require four tosses
>>while
>>>> Bob will require six tosses (try this at home!), even though
>>>> head-tail and head-head have an equal chance of appearing after two
>>>> coin tosses.
>>>
>>> I did try this at home; only I wrote a Python script to do all the
>>> tedious tossing and accounting. This is its output:
>>>
>>>> $ ./cointoss HH HT
>>>>
>>>> H                   T                   HH                  HT
>>>> ----------          ----------          ----------         
>>----------
>>>> 59821 (49.9%)       60079 (50.1%)       6.044               3.990
>>>
>>>
>>> After over a million coin tosses, it takes 6 tosses on average until
>>you
>>> see two heads in a row, but only 4 to see head-tail. Obviously, the
>>> script is attached. Supply the patterns on invocation, as shown
>>above.
>>> Any number of patterns of any length are supported (I think). Well,
>>> strictly positive numbers and lengths :).
>>>
>>> Can someone point me in the direction of the solution to this
>>> counterintuitive probability theory result? Any of a common name for
>>the
>>> property, a mathematical explanation or an intuitive explanation are
>>> much appreciated!
>>>
>>> Anyway, to make up for the off-topicness, let's get slightly
>>on-topic...
>>>
>>> To the OP: Please provide at least a short abstract of the text when
>>you
>>> post a link. That way people can tell from your mail what the text
>>will
>>> be about.
>>>
>>> With regards to the article, I'm surprised by the choice of words in
>>its
>>> title. Other than to draw in more readers, I don't see what place
>the
>>> word "conspiracy" has in it. That's like saying 0 and 1 are
>>conspiring
>>> to be consecutive on the integral number line. Oh no, pretty much
>all
>>> are computers are based on 0's and 1's and now they are conspiring!
>>> Probably against us! Quick, we need neutral numbers without an
>>agenda...
>>> In my opinion, this title really devalues the article. "Three secret
>>> ways to cope with prime conspiracy mathematicians don't want you to
>>know
>>> about" isn't that much further out. Oh, I hope that phrasing doesn't
>>> tickle any spam filters... Ah well.
>>>
>>> Cheers,
>>>
>>> Peter.
>>>
>>> --
>>> I use the GNU Privacy Guard (GnuPG) in combination with Enigmail.
>>> You can send me encrypted mail if you want some privacy.
>>> My key is available at
>><http://digitalbrains.com/2012/openpgp-key-peter>
>>>
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>>
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>
>I've followed the thread for a bit now, but the concept definitely
>brings some things to light, especially for those less
>cryptographically or mathematically inclined. By the basics that I
>know, a 50-50 chance is a 50-50 chance. But as has been pointed out,
>the chance of getting a specific set of results consecutively is
>obviously (according to the data), not 50/50 even though the initial
>probability would imply that. I don't really have much more to add to
>the discussion other than it made me think a bit more about how
>probability and the effect that measuring the probability of
>predetermined sequences within that same set might produce results that
>are contradictory to the initial expectations. Such is the nature of
>these things and I merely found it interesting that the results defied
>the expectation. Which is the essence of discovery and progress.

But to reply directly to the post at hand, inertia would definitely be a factor in a physical coin tossing. Even though the coin only has 2 sides, the amount of initial force used to flip it and all the other external factors affecting it before it landed would play a part, so though the beginning ratio might be 50/50, there are external factors that skew the ratio one way or the other in a true physical coin toss, so on that point, I applaud you for that observation.
-- 
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